Pion Decay is a fundamental process in particle physics, providing key insights into the weak interaction and the Standard Model. This article delves into the process of pion decay, specifically focusing on the decay of a negatively charged pion (π⁻) into a muon (μ⁻) and an anti-muon neutrino (ν̄). This decay is mediated by the charged W boson, a force carrier of the weak interaction. We will explore the Feynman diagram representing this process, the associated Feynman amplitude, and the calculation of the decay rate (Γ).
The Pion Decay Process: π⁻ → μ⁻ + ν̄
The decay of a negative pion (π⁻) into a muon (μ⁻) and an anti-muon neutrino (ν̄) is a classic example of weak interaction. This process occurs because pions are composite particles made of quarks, and are subject to weak decay. The underlying quark process involves the transformation of a down quark within the pion into an up quark, facilitated by the emission of a W⁻ boson. This W⁻ boson subsequently decays into a muon and an anti-muon neutrino.
The Feynman diagram visually represents this interaction:
This diagram illustrates the pion decaying at a vertex, emitting a W⁻ boson which then propagates and decays into a muon and an anti-muon neutrino. The lines represent the particles, and the vertex represents the interaction point.
Feynman Amplitude for Pion Decay
The Feynman amplitude (##mathscr{M}##) is a crucial quantity in particle physics as its square is proportional to the probability of the process occurring. For pion decay, neglecting higher order terms and assuming a massless anti-muon neutrino, the Feynman amplitude is given by:
begin{equation*}
mathscr{M} = frac{g^2}{8 m^2_W}f_{pi} m_{mu} bar{u}_r(vec p) (1-gamma_5)v_s (vec q)
end{equation*}
Where:
- g is the weak coupling constant.
- mW is the mass of the W boson.
- fπ is the pion decay constant, parameterizing the pion’s internal structure.
- mμ is the mass of the muon.
- bar{u}_r(vec p) and v_s (vec q) are Dirac spinors for the outgoing muon and anti-neutrino respectively.
- (1-gamma_5) is the chiral projection operator, reflecting the V-A (vector minus axial-vector) nature of the weak interaction.
Calculating the Decay Rate (Γ)
The decay rate (Γ) represents the probability per unit time that a pion will decay. It is directly related to the square of the Feynman amplitude. The differential decay rate is defined and after performing the necessary calculations, including squaring the amplitude and summing over final state spins, the decay rate for pion decay is expected to be:
begin{equation*}
Gamma = frac{g^4 f_{pi}^2}{256 pi} frac{m^2_{mu}m_{pi}}{m^4_{W}} left( 1 – frac{m^2_{mu}}{m_{pi}^2} right)^2
tag{*}
end{equation*}
Where mπ is the mass of the pion. This formula highlights the dependence of the pion decay rate on fundamental constants, particle masses, and the pion decay constant. The calculation involves trace evaluation in spinor space, utilizing properties of gamma matrices and summing over spins to arrive at this result.
Conclusion
Understanding pion decay is crucial for grasping the intricacies of weak interactions within the Standard Model. The decay rate formula provides a quantitative measure of this process, linking theoretical calculations to measurable physical quantities. Further detailed calculations, involving the trace evaluation of the Feynman amplitude, are necessary to fully verify and understand the origin of this important formula in particle physics.
